Angular displacement θ, angular velocity ω, angular acceleration α: Worked examples

1.5.2 · D3 · Physics › Rotational Mechanics › Angular displacement θ, angular velocity ω, angular accelera

Shuru karne se pehle, ek habit jo tumhe har baar bachayegi:

Scenario matrix

Har rotational-kinematics problem in cells mein se ek hai (ya blend). Neeche har example us cell ke saath tagged hai jise woh cover karta hai.

Cell	Kya cheez use alag banati hai	Kaunsa trap chhupta hai	Example
A. Speed-up from rest	$ω_{0} = 0$ , $α > 0$	rad vs rev bhool jaana	Ex 1
B. Slow-down to rest, no time	$ω, ω_{0}, θ$ pata hai, $α$ chahiye	kaunsi equation (time-free)	Ex 2
C. Reversal	$α$ opposite to $ω$ , motion zero cross karta hai	net vs total angle	Ex 3
D. Angular → linear at a point	$v, a_{t}, a_{c}$ chahiye radius $r$ par	$a_{t}$ vs $a_{c}$ confusion	Ex 4
E. Two radii, one body	same $ω, α$ , alag $v$	kaun kya share karta hai	Ex 5
F. Degenerate: $α = 0$	uniform spin, period/frequency	$α$ equations blindly use nahi kar sakte	Ex 6
G. Real-world word problem	rpm units, "kitne turns"	unit conversion rpm→rad/s	Ex 7
H. Exam twist: coupled wheels	belt/gear do rims ko link karta hai	equal rim speed, equal $ω$ nahi	Ex 8

Example 1 — Cell A: rest se speed-up

Worked example Statement

Ek potter's wheel rest se shuru hota hai aur constant angular acceleration ke $4 s$ baad $ω = 20 rad/s$ tak pahunchta hai. $α$ aur total angle turned nikalo. Kitne full revolutions hain woh?

Forecast: padhne se pehle guess karo — angle $40$ rad se zyada hoga ya kam? (Average speed $20$ ki aadhi hai, $4$ s ke upar...)

$α$ nikalo: $α = \frac{ω - ω _{0}}{t} = \frac{20 - 0}{4} = 5 rad/s^{2} .$ Yeh step kyun? Constant $α$ ke saath definition $α = Δ ω /Δ t$ exact hai — average equals instantaneous, toh yeh bas rise over run hai.
$θ$ nikalo: $θ = ω_{0} t + \frac{1}{2} α t^{2} = 0 + \frac{1}{2} (5) (4)^{2} = 40 rad .$ Yeh step kyun? Hume $ω_{0}, α, t$ pata hai aur $θ$ chahiye — displacement equation exactly yahi use karta hai.
Revolutions mein convert karo: $N = \frac{θ}{2 π} = \frac{40}{2 π} \approx 6.37 rev .$ Yeh step kyun? Ek full turn $2 π$ rad hai; divide karke turns count hote hain.

Verify: Average angular speed $= (0 + 20) /2 = 10 rad/s$ , times $4 s = 40 rad$ . ✓ Match karta hai. Units: $(rad/s) \cdot s = rad$ . ✓

Example 2 — Cell B: rest tak slow down, time nahi diya

Worked example Statement

Ek ceiling fan $ω_{0} = 30 rad/s$ par spin kar raha hai, switch off hota hai aur $θ = 10 rad$ turn karke ruk jaata hai. Uska angular acceleration nikalo.

Forecast: $α$ ka sign — positive hoga ya negative? (Woh slow ho raha hai, aur humne uski spin ko $+$ choose kiya hai...)

Sahi equation choose karo. Hume $ω = 0, ω_{0} = 30, θ = 10$ pata hai, lekin time nahi. Bina $t$ wali equation hai $ω^{2} = ω_{0}^{2} + 2 α θ$ . Yeh step kyun? $ω = ω_{0} + α t$ tak pahunchna humein $t$ invent karne par majboor karega. Knowns ko us equation se match karo jo unknown ko chhodti hai.
Solve karo: $0 = 3 0^{2} + 2 α (10) \Rightarrow 20 α = - 900 \Rightarrow α = - 45 rad/s^{2} .$ Yeh step kyun? Ek baar equation choose ho jaaye toh pure algebra hai.

Verify: Minus sign sahi hai — $α$ (positive) spin ke opposite hai, jo exactly yahi hai jo "slowing down" ka matlab hai hamare sign rule mein. Sanity check: $∣ α ∣$ bada hai kyunki bahut saari speed ( $30$ ) ek chhote angle ( $10$ rad) mein khatam hoti hai. ✓

Example 3 — Cell C: reversal (net vs total angle)

Worked example Statement

Ek disc $ω_{0} = + 6 rad/s$ (counter-clockwise) par spin karti hai jabki constant $α = - 3 rad/s^{2}$ act karta hai. Nikalo (a) momentarily rukne ka time, (b) $t = 4 s$ par uska net angular displacement, aur (c) un $4 s$ mein actually swept total angle.

Forecast: $t = 4 s$ par disc forward spin kar rahi hai, ruki hai, ya backward? Kya net aur total angle equal honge?

Rukne ka time: $ω = ω_{0} + α t$ mein $ω = 0$ set karo: $0 = 6 + (- 3) t \Rightarrow t = 2 s .$ Yeh step kyun? "Momentarily stops" ka matlab $ω = 0$ hai; velocity equation $ω$ ko $t$ se jodhti hai.
$4$ s par net displacement: $θ = ω_{0} t + \frac{1}{2} α t^{2} = 6 (4) + \frac{1}{2} (- 3) (4)^{2} = 24 - 24 = 0 rad .$ Yeh step kyun? Displacement equation signed position deta hai — forward aur backward sweeps cancel ho jaate hain.
Total angle swept. Forward phase ( $0$ to $2$ s): $θ_{1} = 6 (2) + \frac{1}{2} (- 3) (2)^{2} = 12 - 6 = 6 rad$ . Backward phase ( $2$ to $4$ s): symmetry se yeh $6 rad$ doosri taraf sweep karta hai. Total distance $= 6 + 6 = 12 rad .$ Yeh step kyun? Distance direction ignore karta hai — tumhe turning point $t = 2$ s par split karna hoga aur magnitudes add karni hongi, kabhi subtract nahi.

Verify: $t = 4$ par: $ω = 6 + (- 3) (4) = - 6 rad/s$ — exactly start se reverse, $θ_{net} = 0$ ko mirror karta hai (woh wahan waapas hai jahan shuru hua tha, ab backward spin kar raha hai). ✓ Net $0 \neq =$ total $12$ : classic reversal signature. ✓

Example 4 — Cell D: angular ko ek point par linear mein convert karna

Worked example Statement

Example 1 ke potter's wheel ( $ω = 20 rad/s$ , $α = 5 rad/s^{2}$ ) ko radius $r = 0.30 m$ par lo, us instant par jab woh $20 rad/s$ tak pahunchta hai. Rim ki linear speed $v$ , tangential acceleration $a_{t}$ , centripetal acceleration $a_{c}$ , aur total linear acceleration magnitude $a$ nikalo.

Forecast: high speed par kaunsa bada hoga — $a_{t}$ ya $a_{c}$ ?

Linear speed: $v = r ω = 0.30 \times 20 = 6 m/s .$ Yeh step kyun? $v = r ω$ turning rate se rim point ke actual travel speed tak ka bridge hai.
Tangential acceleration: $a_{t} = r α = 0.30 \times 5 = 1.5 m/s^{2} .$ Yeh step kyun? $a_{t}$ speed-changing part hai; yeh $α$ se aata hai, jo changing spin rate hai.
Centripetal acceleration: $a_{c} = ω^{2} r = 2 0^{2} \times 0.30 = 120 m/s^{2} .$ Yeh step kyun? Constant speed par bhi ek circle par ek point turning keep karne ke liye inward accelerate karta hai; woh $a_{c}$ hai, sirf direction-changing.
Total: $a = a_{t}^{2} + a_{c}^{2} = 1. 5^{2} + 12 0^{2} \approx 120.0 m/s^{2} .$ Yeh step kyun? $a_{t}$ (path ke along) aur $a_{c}$ (inward) perpendicular hain, toh Pythagoras se combine karo.

Verify: $a_{c} ≫ a_{t}$ — high $ω$ par turning dominate karta hai, toh total almost pure centripetal hai, jaise expected. Units: $(rad/s)^{2} \cdot m = m/s^{2}$ (radian dimensionless hota hai). ✓

Example 5 — Cell E: ek rigid body par do radii

Worked example Statement

Usi wheel par ( $ω = 20 rad/s$ , $α = 5 rad/s^{2}$ ), ek bug hub $r_{1} = 0.10 m$ par baitha hai aur clay ka ek chip rim $r_{2} = 0.30 m$ par. Unke $ω$ , linear speeds, aur tangential accelerations compare karo.

Forecast: $ω, v, a_{t}$ mein se kaunsa same hai dono ke liye?

Angular quantities: dono $ω = 20 rad/s$ aur $α = 5 rad/s^{2}$ share karte hain. Yeh step kyun? Ek rigid body ek piece ki tarah ghoomta hai — har point same time mein same angle sweep karta hai, toh $θ, ω, α$ sabke liye common hain.
Linear speeds: $v_{1} = r_{1} ω = 0.10 \times 20 = 2 m/s$ ; $v_{2} = r_{2} ω = 0.30 \times 20 = 6 m/s .$ Yeh step kyun? $v = r ω$ : bahar zyada matlab faster, chahe identical $ω$ ho.
Tangential accelerations: $a_{t 1} = r_{1} α = 0.5 m/s^{2}$ ; $a_{t 2} = r_{2} α = 1.5 m/s^{2} .$ Yeh step kyun? $a_{t} = r α$ usi tarah $r$ ke saath scale karta hai.

Verify: Ratios: $v_{2} / v_{1} = 6/2 = 3 = r_{2} / r_{1} = 0.30/0.10$ . ✓ Angular quantities exactly match kiye; sirf linear ones radius ke saath scaled hue. Yahi wajah hai ki humne angular variables invent kiye. ✓

Example 6 — Cell F: degenerate case $α = 0$ (uniform spin)

Worked example Statement

Ek hard drive platter steadily spin karta hai (speed-up nahi) $ω = 7200$ rpm (revolutions per minute) par. Uska period $T$ , frequency $f$ , aur angular velocity rad/s mein nikalo. Uska angular acceleration kya hai?

Forecast: kya hum $θ = ω_{0} t + \frac{1}{2} α t^{2}$ yahan use kar sakte hain? $α$ kya hai?

Angular acceleration zero hai: "steadily spins" ⇒ $ω$ constant ⇒ $α = 0$ . Yeh step kyun? Teen kinematic equations abhi bhi hold karti hain, lekin $α = 0$ ke saath woh $ω = const$ aur $θ = ω t$ mein collapse ho jaati hain. $\frac{1}{2} α t^{2}$ term galat mat use karo — woh vanish ho jaata hai.
Frequency: $f = 7200 rev/min = 7200/60 = 120 Hz (rev/s) .$ Yeh step kyun? Frequency turns per second hai; rpm ko $60$ se divide karo.
Period: $T = 1/ f = 1/120 \approx 0.00833 s .$ Yeh step kyun? Period seconds per turn hai — frequency ka reciprocal.
Angular velocity: $ω = 2 π f = 2 π (120) \approx 753.98 rad/s .$ Yeh step kyun? Har turn $2 π$ rad hai; $ω = 2 π / T = 2 π f$ .

Verify: $ω \cdot T = 753.98 \times 0.00833 \approx 6.283 \approx 2 π$ — har period mein ek full turn, jaisa hona chahiye. ✓ Units: $rev/s \to \times 2 π rad/rev = rad/s$ . ✓

Example 7 — Cell G: real-world word problem (rpm, "kitna time")

Worked example Statement

Ek car engine $800 rpm$ par idle karta hai aur constant angular acceleration par $2.5 s$ mein $3000 rpm$ tak rev up hota hai. $α$ (rad/s² mein) aur is rev-up ke dauran crankshaft ke revolutions ki number nikalo.

Forecast: $2.5$ s mein $60$ se zyada revolutions honge ya kam?

Dono speeds ko rad/s mein convert karo: $ω_{0} = 800 \cdot \frac{2 π}{60} \approx 83.78 rad/s$ , $ω = 3000 \cdot \frac{2 π}{60} \approx 314.16 rad/s .$ Yeh step kyun? Har kinematics formula ko rad/s chahiye; rpm pehle convert karna zaroori hai ( $\times 2 π /60$ ).
Angular acceleration: $α = \frac{ω - ω _{0}}{t} = \frac{314.16 - 83.78}{2.5} \approx 92.15 rad/s^{2} .$ Yeh step kyun? Constant $α$ ⇒ angular velocity ka rise over run.
Angle turned: average speed × time use karo: $θ = \frac{ω _{0} + ω}{2} t = \frac{83.78 + 314.16}{2} (2.5) \approx 497.4 rad .$ Yeh step kyun? Constant $α$ ke saath, average angular speed midpoint hai; yeh $α$ ko dobara plug karne se bachata hai.
Revolutions: $N = θ /2 π \approx 497.4/6.283 \approx 79.2 rev .$ Yeh step kyun? Rad ko turns mein convert karo.

Verify: $θ$ ko $θ = ω_{0} t + \frac{1}{2} α t^{2} = 83.78 (2.5) + \frac{1}{2} (92.15) (2.5)^{2} \approx 209.4 + 288.0 = 497.4 rad$ se cross-check karo. ✓ Dono methods agree karte hain. ✓

Example 8 — Cell H: exam twist, belt se linked do wheels

Worked example Statement

Ek chhoti pulley radius $r_{A} = 0.05 m$ ek badi pulley radius $r_{B} = 0.20 m$ ko non-slipping belt ke through drive karti hai. Chhoti pulley $ω_{A} = 40 rad/s$ par spin karti hai. Belt speed, badi pulley ki $ω_{B}$ , aur unke revolution rates ka ratio nikalo.

Forecast: kaunsi pulley zyada tez spin karti hai (bada $ω$ ) — chhoti ya badi?

Belt speed = driver ki rim speed: $v = r_{A} ω_{A} = 0.05 \times 40 = 2 m/s .$ Yeh step kyun? Ek non-slipping belt us rim ki linear speed par chalti hai jise woh wrap karti hai — yeh shared quantity hai, $ω$ nahi.
Same belt pulley B par: $v = r_{B} ω_{B} \Rightarrow ω_{B} = \frac{v}{r _{B}} = \frac{2}{0.20} = 10 rad/s .$ Yeh step kyun? Belt ki linear speed dono rims mein common hai; B ke liye $v = r ω$ invert karo.
Spins ka ratio: $\frac{ω _{A}}{ω _{B}} = \frac{40}{10} = 4 = \frac{r _{B}}{r _{A}} = \frac{0.20}{0.05} .$ Yeh step kyun? Equal rim speed force karta hai $ω \propto 1/ r$ — chhota wheel faster spin karta hai.

Verify: B se belt speed: $r_{B} ω_{B} = 0.20 \times 10 = 2 m/s$ ✓ driver se match karta hai. Chhoti pulley $4 \times$ faster spin karti hai, apne $4 \times$ -chhote radius ke inverse mein. ✓

Recall Cells par quick self-test

"Know $ω, ω_{0}, θ$ , want $α$ , no time" ke liye kaunsi equation? ::: Time-free: $ω^{2} = ω_{0}^{2} + 2 α θ$ . Reversal ke baad, net displacement total angle swept ke equal hota hai? ::: Nahi — net signed hota hai aur cancel ho sakta hai; total har phase ke magnitudes add karta hai, stop time par split karke. Ek platter par alag radii par do clay chips: woh kya share karte hain? ::: $θ, ω, α$ (same); $v, a_{t}, a_{c}$ differ karte hain, $r$ ke saath scale hote hue. Belt-linked pulleys kya share karte hain? ::: Rim (linear) speed $v$ , toh $r_{A} ω_{A} = r_{B} ω_{B}$ ; $ω$ nahi. $3000$ rpm ko rad/s mein convert karo. ::: $3000 \times 2 π /60 \approx 314.16$ rad/s.

Connections

Parent topic (Hinglish) — woh definitions aur derivations jin par yeh examples practice karate hain.
Uniform Circular Motion — Cell F ( $α = 0$ ) exactly yahi case hai.
Centripetal Acceleration and Force — Example 4 mein use kiya gaya $a_{c} = ω^{2} r$ .
Linear Kinematics Equations — same algebra, bas $x, v, a \to θ, ω, α$ swap karo.
Rolling Motion — Cell H ka $v = r ω$ logic zameen par rolling wheel tak extend hota hai.
Torque and Angular Acceleration — in speed-up examples mein $α$ ko cause kya karta hai.
Moment of Inertia — rotational "mass" us $α$ ke peeche.